/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 Factor the polynomial. $$x^{3}... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 20

Factor the polynomial. $$x^{3}-25 x$$

Short Answer

Expert verified
The polynomial factors to \(x(x + 5)(x - 5)\).

Step by step solution

01

Identify Common Factors

The polynomial is \(x^3 - 25x\). The first step is to look for any common factors in the terms. Here, each term has an \(x\) in common, so factor out \(x\). This gives us \(x(x^2 - 25)\).
02

Recognize the Difference of Squares

Examine the expression \(x^2 - 25\). Notice it follows the difference of squares form \(a^2 - b^2\), where \(a = x\) and \(b = 5\). Use the formula \(a^2 - b^2 = (a+b)(a-b)\).
03

Apply Difference of Squares Formula

Apply the difference of squares formula to \(x^2 - 25\). This becomes \((x + 5)(x - 5)\).
04

Write the Fully Factored Expression

Substitute \(x^2 - 25\) from Step 3 back into the expression from Step 1. The fully factored form is \(x(x + 5)(x - 5)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common factors
When factoring polynomials, the first step is often looking for any common factors among the terms. This is crucial because factoring out common factors simplifies the expression, making further steps more manageable.
Consider the polynomial provided: \(x^3 - 25x\). Each term here contains an \(x\). When terms both include \(x\), we can factor it out, just like taking out a common piece from each puzzle piece. This simplifies the problem to \(x(x^2 - 25)\).
Finding common factors:
  • Determine if any numbers or variables repeat across all terms.
  • Factor out the greatest common factor (GCF) from the polynomial.
  • Rewriting the polynomial with the common factor outside the parentheses.

This step is akin to preparing your ingredients before cooking, setting you up for straightforward further factoring.
Difference of squares
The difference of squares is a specific form within polynomials that can simplify your work greatly. This form occurs when you have two terms that are both perfect squares separated by a minus sign. The general formula for factoring a difference of squares is: \[a^2 - b^2 = (a+b)(a-b)\]
In our exercise, the expression \(x^2 - 25\) exemplifies this. Here, \(x^2\) is already a perfect square, and \(25\) is \(5^2\). Recognizing this allows you to apply the difference of squares formula directly here.
Key features:
  • Term one and term two must both be perfect squares.
  • They are separated by a subtraction sign.
  • The result will be a multiplication of sum and difference: \((a+b)(a-b)\).

This technique can substantially decrease complexity, helping you break down a polynomial into simpler linear factors.
Factoring techniques
Different factoring techniques can be applied based on the structure of the polynomial you're working with. For the polynomial \(x^3 - 25x\), we've successfully used two key techniques: factoring out common factors and applying the difference of squares method.
When approaching any factoring problem, consider the type of polynomial you're dealing with:
  • Monomial: Often requires finding GCF among terms.
  • Binomial: Look for patterns such as difference of squares or perfect square trinomials.
  • Trinomial: May involve pairing methods or using the quadratic formula.

By combining these strategies, the original complex expression is reduced to a simpler one, making solutions easier to find and verify. Thus, the expression \(x^3 - 25x\) eventually simplifies to \(x(x + 5)(x - 5)\), demonstrating the power and efficiency of factoring techniques.

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Most popular questions from this chapter

The temperature \(T\) within a cloud at height \(h\) (in feet) above the cloud base can be approximated using the equation \(T=B-\left(\frac{3}{1000}\right) h,\) where \(B\) is the temperature of the cloud at its base. Determine the temperature at \(10,000\) feet in a cloud with a base temperature of \(55^{\circ} \mathrm{F}\) and a base height of 4000 feet. Note: For an interesting application involving the three preceding exercises, see Exercise 14 in the Discussion Exercises at the end of the chapter.

Simplify the expression, and rationalize the denominator when appropriate. \(\sqrt{9 x^{-4} y^{6}}\)

Replace the symbol \(\square\) with elther \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. \(\sqrt{a^{r}} \square(\sqrt{a})\)

Archeologists can determine the height of a human without having a complete skeleton. If an archeologist finds only a humerus, then the height of the individual can be determined by using a simple linear relationship. (The humerus is the bone between the shoulder and the elbow.) For a female, if \(x\) is the length of the humerus (in centimeters), then her height \(h\) (in centimeters) can be determined using the formula \(h=65+3.14 x .\) For a male, \(h=73.6+3.0 x\) should be used. (a) A female skeleton having a 30 -centimeter humerus is found. Find the woman's height at death. (b) A person's height will typically decrease by 0.06 centimeter each year after age \(30 .\) A complete male skeleton is found. The humerus is 34 centimeters, and the man's height was 174 centimeters. Determine his approximate age at death.

Simplify the expression, assuming \(x\) and \(y\) may be negative. $$\sqrt{x^{4} y^{10}}$$
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